Nilpotent Infinitesimals and Synthetic Differential Geometry in Classical Logic

  • Paolo Giordano


An extension of ℝ (and *ℝ) with nilpotent infinitesimals (e.g. h ≠ 0 but h2 = 0) is presented in order to obtain results similar to KockLawvere’s synthetic differential geometry [3], but in a classical and not intuitionistic context. The same extension can be used to add new infinitesimal points to spaces similar to Chen’s ones [1]. In the category of extended spaces we can develop differential geometry not only of usual manifolds but also of infinite dimensional spaces, without coordinates and with a strong geometric intuition, that is in a way that we will call “synthetic”.


Synthetic differential geometry Differential manifolds foundations 


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  1. [1]
    K.T. Chen, On differentiable spaces, Categories in continuum physics, Lecture Notes in Mathematics 1174, Springer-Verlag (1982), 38–42.Google Scholar
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    A. Frölicher, A. Kriegl, Linear spaces and differentation theory, John Wiley & sons (1988).Google Scholar
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    A. Kock, Synthetic Differential Geometry, London Math. Soc. Lect. Note Series 51, Cambrige Univ. Press (1981).Google Scholar
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    R. Lavendhomme, Lecons de géométrie différentielle synthétique naïve, Ciaco (1987).Google Scholar
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    F.W. Lawvere, Categorical Dynamics, Topos theoretic Methods in Geometry, Aarhus, Various Publications Series no. 30 (1979), 1–28.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Paolo Giordano
    • 1
  1. 1.Accademia di architetturaUniversità della Svizzera italianaMendrisioSwitzerland

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